Available at www.jmilne.org/math/

Erratum

p.16, Theorem 2.1. In the first two paragraphs of the proof of (b), replace $\mathcal{O}_k$ with $\mathcal{O}_{k,\mathfrak{P}}$.

The point is that, if $E$ and $k$ are number fields, then in general an isomorphism $E\otimes_{\mathbb{Q}}k\rightarrow k^n$ doesn't induce an isomorphism $\mathcal{O}_{E}\otimes_{\mathbb{Z}}\mathcal{O}_{k}\rightarrow \mathcal{O}_{k}^{n}$. Suppose, for example, that $\mathcal{O}_{E}=\mathbb{Z}[X]/(f(X))$ with $f(X)$ monic. Then the first isomorphism exists if $f(X)$ splits in $k[X]$, but for the second to exist we need the roots to be distinct modulo every prime ideal in $\mathcal{O}_{k}$. From a different perspective, $\mathcal{O}_{k}^{n}$ is étale over $\mathcal{O}_k$, and so the existence of an isomorphism would imply that $\mathcal{O}_{E}$ is étale (i.e., unramified) over $\mathbb{Z}$ (by descent).